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Complexity As a Homeomorphism Invariant for Tiling Spaces Tome 67, No 2 (2017), P R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Antoine JULIEN Complexity as a homeomorphism invariant for tiling spaces Tome 67, no 2 (2017), p. 539-577. <http://aif.cedram.org/item?id=AIF_2017__67_2_539_0> © Association des Annales de l’institut Fourier, 2017, Certains droits réservés. Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 3.0 FRANCE. http://creativecommons.org/licenses/by-nd/3.0/fr/ L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 67, 2 (2017) 539-577 COMPLEXITY AS A HOMEOMORPHISM INVARIANT FOR TILING SPACES by Antoine JULIEN (*) Abstract. — It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complex- ity functions are asymptotically equivalent in a certain sense (which implies, if the complexity is polynomial, that the exponent of the leading term is preserved by homeomorphism). This theorem can be reworded in terms of d-dimensional d infinite words: if two Z -subshifts (with the same conditions as above) are flow equivalent, their complexity functions are equivalent up to rescaling. An analogous theorem is stated for the repetitivity function, which is a quantitative measure of the recurrence of orbits in the tiling space. Behind this result is the fact that any homeomorphism between tiling spaces is described by a so-called “shape deforma- tion”. In the last section, we use this observation to show that a certain cohomology group is an invariant of homeomorphisms between tiling spaces up to topological conjugacy. Résumé. — Il est prouvé dans cet article que deux pavages apériodiques et répétitifs dont les espaces de pavages sont homéomorphes ont des fonctions de complexité asymptotiquement équivalentes en un certain sens. Cela implique que lorsque les fonctions de complexité croissent polynomialement, l’exposant du terme dominant est préservé par homéomorphisme. Ce théorème peut s’énoncer en termes d de mots infinis d-dimensionels: si deux sous-décalages indexés par Z (avec les mêmes conditions) sont « flot-équivalents » (c’est-à-dire que leurs suspensions sont homéomorphes), alors leurs fonctions de complexité sont équivalentes à changement d’échelle près. Un théorème analogue peut être énoncé pour la fonction de répé- titivité, qui donne une mesure quantitative de la vitesse de récurrence des orbites dans l’espace de pavages. De manière sous-jacente, se trouve le fait que tout ho- méomorphisme entre espaces de pavages est induit par une déformation des tuiles. Dans la dernière section, on utilise cette observation pour montrer qu’un certain groupe de cohomologie fournit un invariant des homéomorphismes entre espaces de pavages à conjugation près. Keywords: aperiodic tilings, complexity, repetitivity, flow-equivalence, orbit-equivalence. Math. classification: 37B50, 37B10. (*) Part of the early work on this topic was done at the University of Victoria, under the supervision of Ian Putnam, partially funded by the Pacific Institute for the Mathematical Sciences. 540 Antoine JULIEN 1. Introduction This article is concerned with the study of aperiodic tilings. These tilings, such as the well-known Penrose tilings, provide good models for quasicrys- tals in physics. The tilings in which we are interested display two a priori antagonistic properties: a highly ordered structure (in the form of uniform repetition of patches), and aperiodicity. A tiling is a covering of the plane by geometric shapes (tiles), with no holes and no overlap. In this paper, tilings are assumed to be made using finitely many tile types up to translation. Furthermore, it is assumed that any two tiles can be fitted together in only finitely many ways (for example it could be assumed that tiles meet full-face to full-face in the case of polytopal tiles). This last condition is known as finite local complexity. Finally, tilings we are dealing with are assumed to be repetitive in the sense that any finite patch of a given tiling repeats within bounded distance of any point in the tiling (these definitions are given with more precision and quantifiers in Section2). An aperiodic tiling with the above properties has a topological compact space associated with it. It consists of all tilings which are indistinguish- able from it at a local scale, and it supports an Rd-action (for tilings of dimension d), induced by translations. It is possible to gain a better un- derstanding of such tiling spaces by using an analogy with subshifts: a d multi-dimensional infinite word w ∈ AZ has a subshift associated with it. d It is the closure in AZ of the Zd-orbit of w under the shift (or translation) map. The word w can also be interpreted as a tiling by cubes, the colours of which are indexed by A. The tiling space of this tiling by cubes then cor- responds exactly to the suspension of the subshift (which for d = 1 is also known as the mapping torus of the shift map). This suspension contains the subshift as a closed subset, and supports an Rd-action which extends the Zd-action on the subshift. We see here that if a subshift is minimal (which corresponds exactly to the condition cited above on repetition of patches), all elements in the subshifts have the same language, i.e. the same set of finite subwords. In the same way, if a tiling space is minimal, it make more sense to study the space rather than to particularise an arbitrarily chosen tiling. This philosophy of studying the space rather than the tiling (or the sub- shift rather than the word) is far from new, and has been quite successful. One can specifically cite the gap-labeling theorems, which relate the topo- logical invariants of the space on the one hand (the ordered K0-group), and the gaps in the spectrum of a certain Schrödinger operator with aperiodic ANNALES DE L’INSTITUT FOURIER COMPLEXITY AND HOMEOMORPHISMS BETWEEN TILING SPACES 541 potential on the other hand. Any attempt to give complete references for this problem would be unfair; we will therefore give a very incomplete list of references in the form of the papers [4, 13,3] and their references. A natural question, however, which results directly from this “space-over- tiling” approach is the following: whenever a relation is established between two tiling spaces, to what extent does it translate back in terms of tilings? In this paper, we start with two tiling spaces which are equivalent in the realm of topological spaces, i.e. homeomorphic. We then investigate what are the consequences on the underlying tilings. The main results state that whenever two tiling spaces are homeomorphic, their respective complexity functions, as well as their repetitivity functions are equivalent in some sense. Given a tiling (or a word), the complexity function is a map r 7→ p(r), where p(r) counts the number of distinct patches of size r, up to translation. This function was first studied in the framework of symbolic dynamics (one- dimensional subshifts). In their seminal paper, Morse and Hedlund [20] define p(n) as the number of subwords of size n in a given bi-infinite word. This function provides a good measure of order and disorder in a word: if p(n) doesn’t grow at least linearly, then the word is periodic. This result was generalized outside of the symbolic setting, in higher dimension: if the complexity function associated with a tiling doesn’t grow at least linarly, then the tiling is completely periodic i.e. it has d independent periods. One can refer to Lagarias–Pleasants [18] for a proof of this result in the setting of Delaunay sets of Rd. The first main theorem of this article describes how the complexity func- tion is preserved whenever two tiling spaces are homeomorphic. Theorem (Theorem 3.1).— If h :Ω −→ Ω0 is a homeomorphism be- tween two tiling spaces of aperiodic and repetitive tilings with finite local complexity, then the associated complexity functions p and p0 satisfy the following inequalities: 0 cp(mr) 6 p (r) 6 Cp(Mr), for some constants c, C, m, M > 0 and all r big enough. A few remarks: (1) If the functions p and p0 are at least linear (which is the case for non-periodic tilings, by the Morse–Hedlund theorem), then c and C can be taken equal to 1 (up to adjusting the values of m and M); (2) On the other hand, if the functions p and p0 grow at most polyno- mially, then m and M can be chosen equal to 1 (up to adjusting c TOME 67 (2017), FASCICULE 2 542 Antoine JULIEN and C). In particular, if p grows like a polynomial, then so does p0, and the exponent of the leading term is the same. Corollary 1.1. — Let T be an aperiodic, repetitive tiling with finite local complexity. Assume that its complexity function grows asymptotically like rα, up to a multiplicative constant. Then the exponent α is a topological invariant of the tiling space associated with T . The repetitivity function was also introduced by Morse and Hedlund [20] in the setting of one-dimensional symbolic dynamics. The repetitivity func- tion for a tiling T is defined as follows: for r > 0, R(r) is the infimum of all numbers c which satisfy that any patch of radius c in T contains a copy of all patches of radius r which appear somewhere in T .
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